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Recoverable Creep Deformation and Transient Local Stress Concentration Due to Heterogeneous Grain-Boundary Diffusion and Sliding in Polycrystalline Solids

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Recoverable Creep Deformation and Transient Local Stress Concentration Due to Heterogeneous Grain-Boundary Diffusion and Sliding in Polycrystalline Solids ARTICLE IN PRESS Journal of the Mechanics and Physics of Solids 56 (2008) 1460–1483 www.elsevier.com/locate/jmps Recoverable creep deformation and transient local stress concentration due to heterogeneous grain-boundary diffusion and sliding in polycrystalline solids Yujie Wei, Allan F. BowerÃ, Huajian GaoÃÃ Division of Engineering, Brown University, Box D, Providence, RI 02912, USA Received 8 June 2007; received in revised form 14 August 2007; accepted 15 August 2007 Abstract Numerical simulations are used to investigate the influence of heterogeneity in grain-boundary diffusivity and sliding resistance on the creep response of a polycrystal. We model a polycrystal as a two-dimensional assembly of elastic grains, separated by sharp grain boundaries. The crystal deforms plastically by stress driven mass transport along the grain boundaries, together with grain-boundary sliding. Heterogeneity is idealized by assigning each grain boundary one of two possible values of diffusivity and sliding viscosity. We compute steady state and transient creep rates as functions of the diffusivity mismatch and relative fractions of grain boundaries with fast and slow diffusion. In addition, our results show that under transient conditions, flux divergences develop at the intersection between grain boundaries with fast and slow diffusivity, which generate high local stress concentrations. The stress concentrations develop at a rate determined by the fast diffusion coefficient, and subsequently relax at a rate determined by the slow diffusion coefficient. The influence of the mismatch in diffusion coefficient, loading conditions, and material properties on the magnitude of this stress concentration is investigated in detail using a simple model problem with a planar grain boundary. The strain energy associated with these stress concentrations also makes a small fraction of the plastic strain due to diffusion and sliding recoverable on unloading. We discuss the implications of these results for conventional polycrystalline solids at high temperatures and for nanostructured materials where grain-boundary diffusion becomes one of the primary inelastic deformation mechanisms even at room temperature. r 2007 Published by Elsevier Ltd. Keywords: Recoverable creep deformation; Grain-boundary diffusion; Grain-boundary sliding; Transient stress concentration 1. Introduction Grain-boundary (GB) diffusion and sliding are the dominant mechanisms of plastic deformation in polycrystalline metals and ceramics at high homologous temperatures. They also contribute to room- temperature plastic flow in nanocrystalline materials, where the fine grain size tends to suppress plastic flow by ÃCorresponding author. ÃÃAlso corresponding author. E-mail addresses: [email protected] (A.F. Bower), [email protected] (H. Gao). 0022-5096/$ - see front matter r 2007 Published by Elsevier Ltd. doi:10.1016/j.jmps.2007.08.007 ARTICLE IN PRESS Y. Wei et al. / J. Mech. Phys. Solids 56 (2008) 1460–1483 1461 dislocation motion while the high density of GBs accelerates diffusional creep. In addition, GB diffusion plays an important role in stress generation and relaxation in polycrystalline thin films. GB diffusional creep in macroscopic polycrystals has been extensively studied. Nabarro (1948) and Herring (1950) first pointed out that self-diffusion can cause crystals to change shapes and induce macroscopic plastic deformation at elevated temperatures. Coble (1963) studied the dependence of macroscopic strain rates _ on grain size l and applied tensile stress s due to collective GB diffusion and showed that dDO s _ ¼ a 3 , (1) kBT l where a is a geometrical constant, D is the GB diffusivity and d is the thickness of a layer in which interface diffusion is supposed to take place; kB, T and O are the Boltzman constant, the absolute temperature and the atomic volume, respectively. Ashby and Verrall (1973) considered a geometrical model of creep flow taking into account strains due to GB diffusion and sliding, to reflect the fact that both GB diffusion and sliding plays an increasingly important role in the inelastic deformation of polycrystalline solids as the temperature rises or the grain size decreases. More recent work has extended the original Coble model by accounting for phenomena such as interface reactions (Ashby, 1969, 1972; Arzt et al., 1983; Cocks, 1992). In general, these models predict a nonlinear relationship between stress and strain rate. Numerical methods have been developed to study GB diffusion and associated mechanical behavior in polycrystalline solids. Needleman and Rice (1980) have first applied the finite-element method in modeling GB diffusion and creep. Pan and Cocks (1993) developed finite-element formulations for modeling GB diffusion in arbitrary networks of grains with straight GBs. They applied the numerical techniques, combined with a time integration algorithm, to study microstructure evolution during superplastic deformation in polycrystalline materials. Bower and Wininger (2004) extended the work of Pan and Cocks (1993) by using an advancing front algorithm to generate a sequence of adaptive, evolving finite-element meshes to solve the evolution of two-dimensional geometries. Sethian and Wilkening (2003) and Wilkening et al. (2004a,b) have used techniques from semigroup theory to study mass transport in microelectronic circuits due to electromigration and GB diffusion. GB diffusion is also an important stress relaxation mechanism in thin films based on experimental observations (Thouless et al., 1996; Kobrinsky and Thompson, 1998) and theories (Gao et al., 1999; Weiss et al., 2001; Guduru et al., 2003). Meanwhile, it has been observed to act in combination with other deformation mechanisms in thin films. For example, recent studies on the mechanical behaviors of polycrystalline thin films on substrates (Gao et al., 1999) have indicated that constrained GB diffusion1 induces crack-like singular stress fields which leads to novel dislocation mechanisms that are driven by locally induced, rather than globally applied, stresses. In situ TEM experiments (Balk et al., 2003), atomistic simulations (Buehler et al., 2003) and discrete dislocation simulations (Hartmaier et al., 2005) have shown that such constrained GB diffusion and the associated dislocation mechanisms dominate plastic deformation mechanisms in unpassivated films thinner than a few hundred nanometers. These studies are calling for broader investigations on the general importance of constrained GB diffusion mechanisms in polycrystalline materials. Coble-type GB diffusional creep, together with GB sliding accommodated by GB diffusion, have been considered as major deformation mechanisms in nanocrystalline materials by many authors (e.g., Gleiter, 1989). Although diffusion-controlled processes are typically activated only at high homologous temperatures, Yamakov et al. (2002) have argued, based on their molecular dynamics simulations, that Coble-creep should be a dominating mechanism in nanocrystalline materials even at low homologous temperatures. An important feature of plastic deformation controlled by GB diffusion is that it gives rise to a strain-rate sensitivity on the order of 1. Although strain-rate sensitivity around 1 is rarely observed in regular tensile and compressive tests for nanocrystalline materials, significantly enhanced strain-rate sensitivity has been reported in both h.c.p. and f.c.c. nc materials. Nc Cu synthesized by Jiang et al. (2006) showed a rate sensitivity about m ¼ 0:104 at room 1In columnar grain structures in thin films on substrates, diffusion tends to generate stress singularities at the ends of GBs that terminate perpendicular to the substrate, if diffusion between thin films and substrates are constrained. These stress singularities may lead to dislocation emission and initiate delamination. GB diffusion in such a situation is termed as ‘constrained GB diffusion’. ARTICLE IN PRESS 1462 Y. Wei et al. / J. Mech. Phys. Solids 56 (2008) 1460–1483 1000 4 × 10-1/s 800 4 × 10-2/s 4 × 10-3/s 600 4 × 10-4/s 4 × 10-5/s Stress (MPa) 400 4 × 10-6/s 200 2 × 10-6/s 10-6/s 0 0 2 4 6 Strain (%) Fig. 1. Stress–strain curves of brush-plated nanocrystalline Cu at different strain rates. Data replotted from Jiang et al. (2006). temperature in the strain rate range of 10À6=sto4Â 10À1=s, Fig. 1. The authors have attributed the high strain-rate sensitivity to the dominant unstructured high-angle GBs in the material. With the current technological trends in continuing miniaturization of structures in electronic devices and materials, a thorough understanding of the inelastic deformation mechanisms by GB diffusion and GB sliding is becoming increasingly important. On the other hand, existing experiments and theories have not been able to fully capture the complex interplay between different deformation mechanisms in GBs. In particular, the following issues are of interest for the present study. Most existing theories have assumed homogeneous diffusivity in GBs. In reality, most material properties are heterogeneous at the scale of individual grains; high angle, relatively unstructured GBs usually have higher diffusivities compared to low angle, structured GBs. GB diffusivity in a polycrystalline solid could differ by several orders of magnitude from one grain to another. Constrained GB diffusion due to heterogeneous diffusivities can affect not only deformation mechanisms in GBs, but also those in grain interiors. Diffusional deformation in GBs is a kinetic process but existing theories are
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